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Asymptotic behavior of solutions of difference equations in Banach spaces

Author(s): Cristóbal González; Antonio Jiménez-Melado
Journal: Proc. Amer. Math. Soc. 128 (2000), 1743-1749.
MSC (1991): Primary 39A10; Secondary 47N99
Posted: February 3, 2000
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Abstract | References | Similar articles | Additional information

Abstract: In this paper we consider the first order difference equation

$\begin{displaymath}\Delta x_n = \sum _{i=0}^\infty a_n^i f(x_{n+i}), \end{displaymath}$

and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation

$\begin{displaymath}\Delta (q_n \Delta x_n) + p_n f(x_n) =0 \end{displaymath}$

to have asymptotically constant solutions.

References:

[D-P87]: A. DROZDOWICZ AND J. POPENDA, Asymptotic Behavior of the Solutions of the Second Order Difference Equation. Proc. Amer. Math. Soc. 99-1 (1987), 135-140. MR 88d:39010
[L-St85]: P. K. LIN AND Y. STERNFELD, Convex Sets with the Lipschitz Fixed Point Property are compact. Proc. Amer. Math. Soc. 93 (1985), 633-639. MR 86c:47074
[P-S94]: J. POPENDA AND E. SCHMEIDEL, On the Asymptotic Behavior of Solutions of Linear Difference Equations. Publications Matemàtiques. 38 (1994), 3-9. MR 95j:39010
[S97]: E. SCHMEIDEL, On the Asymptotic Behaviour of Solutions of Difference Equations. Demonstratio Mathematica. XXX-1 (1997), 193-197. MR 98m:39024

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Additional Information:

Cristóbal González
Affiliation: Departamento de Análisis Matemático, Universidad de Málaga, Fac. Ciencias, 29071 Málaga, Spain
Email: gonzalez@anamat.cie.uma.es

Antonio Jiménez-Melado
Affiliation: Departamento de Análisis Matemático, Universidad de Málaga, Fac. Ciencias, 29071 Málaga, Spain
Email: jimenez@anamat.cie.uma.es

DOI: 10.1090/S0002-9939-00-05490-3
PII: S 0002-9939(00)05490-3
Keywords: Difference equation, asymptotic behavior
Received by editor(s): July 21, 1998
Posted: February 3, 2000
Additional Notes: This research was partially supported by a grant from Ministerio de Educación y Cultura (Spain) PB97-1081, and from La Junta de Andalucía
Communicated by: Hal L. Smith
Copyright of article: Copyright 2000, American Mathematical Society


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